Note that taking opposite category gives rise to a $2$-endofunctor:
$$(-)^{op}:\mathrm{CAT}\rightarrow \mathrm{CAT}$$
on $2$-category of all categories contained in a given universe $\mathcal{U}$. This $2$-endofunctor is covariant on functors and contraviariant on natural transfromations.
Using this $2$-endofunctor you can argue as follows. If $$(\mathcal{C},\otimes,\alpha,\eta,\lambda,I)$$ is a monoidal category, then:
$$(\mathcal{C}^{op},\otimes^{op},\alpha,\eta,\lambda,I)$$
satisfies axioms of monoidal category except for the fact that in Mac Lane's pentagon and identity triangles arrows are reversed. So if you are assuming that $\alpha$, $\eta$, $\lambda$ are natural isomorphisms(which is usually assumed c.f Categories for the working mathematician), then:
$$(\mathcal{C}^{op},\otimes^{op},\alpha^{-1},\eta^{-1},\lambda^{-1},I)$$
is a monoidal category